Conjecture 10.14 — The map (50) is an isomorphism.

Definitions and mathematical statement

The conjecture asserts that the source's map (50) is an isomorphism. Its formula, domain, codomain and algebraic category are not supplied. With placeholders $F_{50}:A_{50}\to B_{50}$, the target is bijectivity together with preservation of the intended structure and a structure-preserving inverse; in a group/module setting this would mean $\ker F_{50}=0$ and $\operatorname{im}F_{50}=B_{50}$. No particular diagram-to-graded-space map is reconstructed.

The attached footnote says that $Y_d$-equivalence is also called $(d-1)$-equivalence, due to Goussarov, in some literature. This index shift must be preserved when recovering the definition behind the map. The source further says the conjecture might reduce to Conjecture 10.8 and the following conjecture. Neither the proposed reduction nor the following conjecture is included, so they are not asserted as an established equivalence or a complete proof strategy.

The supplied summary calls this not obviously resolved and provides no exact modern theorem. The isomorphism claim is retained as an incomplete source statement pending recovery of equation (50) and current primary-literature review.

Short English statement

Prove that the omitted map (50) is an isomorphism, with the source's $Y_d$/$(d-1)$ indexing. The map and proposed conjectural reduction are missing, so no replacement map or established reduction is inferred.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400189$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 10.14, PDF page 151, equation (50) and footnote 35: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29.
• Conjecture 10.8 and an unspecified following conjecture are referenced for a possible reduction; their full statements/reduction are not supplied here. Goussarov is credited for the alternate index convention.